Group actions in symplectic and contact topology

点击翻译按钮获取中文摘要

Symplectic geometry has its roots in classical mechanics. It is the mathematical structure that underlies the equations of motion of celestial bodies, spinning tops, and mechanical linkages. The field has gone through spectacular progress in recent decades, and deep connections have emerged with other fields of mathematics as well as theoretical physics.Karshon's research focuses on aspects of symplectic geometry that involve symmetries. A "baby example" is the two dimensional sphere with its rotational symmetry. A higher dimensional example is the complex projective plane; symplectically, this can be viewed as a four dimensional ball with a two dimensional sphere sewed along its edge. As envisioned by Felix Klein in his 1872 "Erlanger programm", there is great benefit in studying a geometry through its group of symmetries. The full symmetry group of a symplectic space is always infinite dimensional; however, for many important spaces (such as the two-sphere or the complex projective plane), inside this infinite dimensional group one can find compact finite dimensional subgroups (which can be thought of rotations in multiple dimensions). Karshon's research programme involves the study of symplectic spaces through these finite dimensional symmetries.Current projects involve new classification schemes, recovering symmetries from their underlying "quotient diffeology", and relations with geometric quantization.